Both Fans stay on no matter what:

There's a temp switch at the auto trans that may be causing your fan run-on. Parallel to the intake switch, you can test for that switch closed by measuring resistance between the two leads at the intake without crawling under the car. Low or no resistance means the trans switch is the culprit.

My first-hand experience with the fan switch on the intake came a week or so after the engine got a little bath as part of an appearance refreshing session. Took that long for the water to migrate down into the delicate workings of the temp switch and convince them to stay in the closed position. Easy diagnosis-- pull one of the wires off the switch and the symptom disappeared. You can test the intake switch with an ohm meter, where an open switch is OK at normal temps, a closed switch at normal temp is a problem. New switch is the solution.

There's also the option that the trans really is overheated. Trust your sensors, but verify.
'89 S4 Auto, black
Dr Bob

One Fan stays on no matter what:

It is very likely that there is a short circuit in the power stage, causing one fan to always stay on when ignition is on. Repair the power unit on the front bumper.

Both Fans stay don't run no matter what:

Fuses? Power amp? Of the Climate controller next to the passenger seat

One Fans stays off no matter what:

Fuse? Fan broken? A good way to verify this is take off the connector at the power amp. Connect pin 1 to 4 and  4 to 8 on the plug. Don't connect it to the power amp. Both fans run at full power when ignition is on. That verifies the fan operation. That leaves either the power amp (unlikely) of the climate controller. I have also seen a defective climate controller twice causing this symptom.